New York Journal of Mathematics a Note on Lattices of Z-Ideals of F-Rings

New York Journal of Mathematics New York J. Math. 22 (2016) 351–361.

A note on lattices of z-ideals of f-rings

Themba Dube

Abstract. The lattice of z-ideals of the ring C(X) of real-valued continuous functions on a completely regular Hausdorﬀ space X has been shown by Mart´ınezand Zenk to be a complete Heyting algebra with cer- tain properties. We show that these properties are due only to the fact that C(X) is an f-ring with bounded inversion. This we do by studying lattices of algebraic z-ideals of abstract f-rings with bounded inversion.

Contents 1. Introduction 351 2. Preliminaries 352 2.1. Rings and f-rings 352 2.2. Algebraic frames 352 3. Frames of z-ideals of f-rings 353 4. Functoriality emanating from z-ideals 359 References 361

1. Introduction It has been remarked by Subramanian [13] that many results in the ring C(X) of real-valued continuous functions on a completely regular Hausdorﬀ space X are mainly due to the fact that C(X) is an f-ring. Of course C(X) is not just any f-ring; it also has some special properties such as bounded inversion, which means that every element above the identity (relative to its natural partial order) is invertible. It also has the property that the sum of two z-ideals in it is a z-ideal [3]. In the course of this introduction we shall use terms which (although standard) we will recall, but only in subsequent sections. Our aim in this paper is to show that it is because C(X) is an f-ring with bounded inversion that the lattice Cz(X) of z-ideals of C(X) is a normal

Received January 26, 2016. 2010 Mathematics Subject Classiﬁcation. Primary: 06F25; Secondary: 06D22, 13A15, 18A05. Key words and phrases. f-ring, z-ideal, coherent frame, proper map, functor, natural transformation. The author acknowledges funding from the National Research Foundation of South Africa through a research grant with Grant No. 93514.

ISSN 1076-9803/2016 351 352 THEMBA DUBE coherent Yosida frame. This we do by showing that if A is any f-ring with bounded inversion, then the lattice ZId(A) of z-ideals of A (here we mean z-ideals in the algebraic sense) is a normal coherent Yosida frame (Theorems 3.2 and 3.5). We hasten to mention that an ideal of C(X) is a z-ideal algebraically if and only if it is z-ideal in the usual topological sense. We show that ZId(A) is a frame by constructing a nucleus, z: RL(A) → RL(A), on the frame RL(A) of radical `-ideals of A whose image is precisely ZId(A). As it turns out, this nucleus induces a codense coherent map RL(A) → ZId(A), which is dense precisely when the Jacobson radical and the nilradical of A coincide (Proposition 3.10). Furthermore, it leads to a natural transformation RL → ZId (Proposition 4.2).

2. Preliminaries 2.1. Rings and f-rings. All rings (including f-rings) in this paper are commutative with identity 1. We recall that an f-ring is a lattice-ordered ring A which satisﬁes the condition that, for any a, b ∈ A and any c ≥ 0 in A, (a ∧ b)c = (ac) ∧ (bc). By a positive element of an f-ring A we mean an element a ≥ 0, and we set A+ = {a ∈ A | a ≥ 0}. Squares are positive in f-rings. An f-ring has bounded inversion if every element a ≥ 1 is invertible. A ring is semiprime if it has no nonzero nilpotent element. By the Jacobson radical of a ring we mean the intersection of all its maximal ideals. We write Max(A) for the set of all maximal ideals of a ring A. For any a ∈ A, we set M(a) = {M ∈ Max(A) | a ∈ M}. An ideal I of A is a z-ideal if, for any a, b ∈ A, M(a) = M(b) and a ∈ I imply b ∈ I. See [11] for a detailed study of z-ideals in rings. Every maximal ideal is a z-ideal, and every minimal prime ideal is a z-ideal. Intersections of z-ideals are z-ideals, and the Jacobson radical of A, here denoted Jac(A), is a z-ideal contained in every z-ideal. An ideal of C(X) is a z-ideal in the usual sense if and only if it is a z-ideal in the algebraic sense as just deﬁned.

2.2. Algebraic frames. The lattices we shall deal with are of a special kind that includes lattices of open sets of a topological space. They are called frames. To recall, a frame is a complete lattice L such that the distributive law _ _ a ∧ S = {a ∧ x | x ∈ S} A NOTE ON LATTICES OF z-IDEALS OF f-RINGS 353 holds for every a ∈ L and S ⊆ L. We denote by 0 the zero of a frame, and by 1 its unit. A frame homomorphism is a mapping h: L → M between frames that preserves ﬁnite meets (including the unit), and arbitrary joins (including the zero). The resulting category is denoted by Frm. Our references for frames are [5] and [12]. We write k(A) for the set of compact elements of a frame A. If k(A) generates A, in the sense that every element of A is the join of compact elements below it, then A is said to be algebraic. An algebraic frame A is said to have the ﬁnite intersection property (FIP) if a∧b ∈ k(A) for all a, b ∈ k(A). A compact algebraic frame with FIP is called coherent, as is a frame homomorphism φ: A → B between coherent frames that takes compact elements to compact elements. As usual, CohFrm denotes the category of coherent frames with coherent frame homomorphisms. A coherent frame is called a Yosida frame [10] if every compact element in it is a meet of maximal elements.

3. Frames of z-ideals of f-rings As stated in the Introduction, we aim to show in this section that some of the properties of the lattices Cz(X) are mainly due to the fact that C(X) is an f-ring with bounded inversion. Recall that an ideal I of an f-ring A (or `-ring, for that matter) is an `-ideal if, for any a, b ∈ A, |a| ≤ |b| and b ∈ I =⇒ a ∈ I. As remarked in [7], if A is an f-ring with bounded inversion, then every z- ideal in A is an `-ideal. In fact, an f-ring A has bounded inversion precisely when every z-ideal in A is an `-ideal. Indeed, if every z-ideal is an `-ideal, then every maximal ideal is an `-ideal, which is known to be equivalent to having the bounded inversion property. We denote by ZId(A) the lattice of z-ideals of A. Banaschewski shows in [2, Proposition 2.4] that the lattice RL(A) of radical `-ideals of any f-ring A is a coherently normal frame. A frame A is coherently normal if it is coherent, and for each compact c in A, the frame ↓c is normal. We show that for any f-ring A with bounded inversion, ZId(A) is a quotient of RL(A). This we do by deﬁning a nucleus z on RL(A) for which Fix(z) = ZId(A). For a ∈ A, set \ M(a) = {M ∈ Max(A) | a ∈ M}. The convention, as usual, is that if a belongs to no maximal ideal (which is the case precisely when a is invertible) then M(a) is the entire ring. Note that, being an intersection of z-ideals (recall that maximal ideals are z- ideals), M(a) is a z-ideal, and is, in fact, the smallest z-ideal containing a. It is clear that M(a2) = M(a) since our maximal ideals are prime. Also, M(0) = Jac(A). Observe that, in any ring A, if I ⊆ A is an ideal, then I is a z-ideal ⇐⇒ ∀a ∈ I,M(a) ⊆ I. 354 THEMBA DUBE

We need the following lemma, which we state more generally that will be needed for our purposes. Lemma 3.1. Let A be an f-ring with bounded inversion, and let a, b ∈ A. (a) M(a) ∩ M(b) = M(ab). (b) If a ≥ 0 and b ≥ 0, then M(ab) = M(a ∧ b). (c) If a ≥ 0 and b ≥ 0, then M(a) + M(b) ⊆ M(a + b). Proof. (a) Let x ∈ M(a) ∩ M(b). If N is a maximal ideal containing ab, then N contains a or it contains b, by primeness. In either case, x ∈ N since x belongs to every maximal ideal containing a, and to every maximal ideal containing b. Thus, x ∈ M(ab), showing that M(a)∩M(b) ⊆ M(ab). On the other hand, ab belongs to every ideal that contains a, and every ideal that contains b. Therefore ab ∈ M(a) ∩ M(b), and hence M(ab) ⊆ M(a) ∩ M(b) since M(a) ∩ M(b) is a z-ideal. Thus, M(ab) = M(a) ∩ M(b). (b) Since ab = (a∨b)(a∧b), ab ∈ M(a∧b), hence M(ab) ⊆ M(a∧b). Also, 0 ≤ (a ∧ b)2 ≤ ab ∈ M(ab), which implies M(a ∧ b) = M((a ∧ b)2) ⊆ M(ab). The claimed equality follows. (c) Since 0 ≤ a ≤ a + b and M(a + b) is an `-ideal (as it is a z-ideal) containing a + b, we have a ∈ M(a + b). Hence M(a) ⊆ M(a + b). Similarly, M(b) ⊆ M(a + b), whence M(a) + M(b) ⊆ M(a + b). Theorem 3.2. For any f-ring A with bounded inversion, ZId(A) is a frame. In fact, it is a quotient of RL(A). Proof. Deﬁne a map z: RL(A) → RL(A) by _ [ z(I) = {M(a) | a ∈ I} = {M(a) | a ∈ I}. RL(A) To see that the join is the union, observe that the collection {M(a) | a ∈ I} is up-directed (henceforth abbreviated “directed”) because for any u, v ∈ I, u2 + v2 ∈ I and M(u) ∪ M(v) ⊆ M(u2 + v2), as one deduces from Lemma 3.1(c). Clearly, I ⊆ z(I) for any I ∈ RL(A). Observe also that z is order-preserving. Now for any I,J ∈ RL(A), _ _ z(I) ∩ z(J) = M(a) ∧ M(b) a∈I b∈J _ = {M(a) ∩ M(b) | a ∈ I, b ∈ J} _ ≤ {M(c) | c ∈ I ∩ J} by Lemma 3.1(a) = z(I ∩ J). Hence z preserves meets. To show idempotency, we need to show that, for any I ∈ RL(A), z(z(I)) ⊆ z(I). Let u ∈ z(z(I)). Pick v ∈ z(I) such that u ∈ M(v). Since z(I) is a z-ideal — being a directed union of z-ideals — and since v ∈ z(I), it follows that M(v) ⊆ z(I), and hence u ∈ z(I). Consequently z(z(I)) ⊆ z(I), and hence equality. Thus, z is a nucleus. A NOTE ON LATTICES OF z-IDEALS OF f-RINGS 355

We have already commented that each z(I), for I ∈ RL(A), is a z-ideal. On the other hand, if J is a z-ideal, then J ∈ RL(A), and z(J) = J. Therefore the image of z is precisely ZId(A), as desired. Remark 3.3. We mentioned in the Preliminaries that the Jacobson radical is the smallest z-ideal in any ring. This is reconfirmed (in the present√ case) by the fact that 0 = z(0 ), and the zero of RL(A) is 0, the ZId(A) RL(A) √ nilradical of A. Simple calculations show that z( 0) = Jac(A). Remark 3.4. We could also have defined the nucleus z on RId(A), the frame of radical ideals of A [1], to arrive at the conclusion that ZId(A) is a quotient of RId(A). Of course the question of whether RId(A) = RL(A) for f-rings with bounded inversion is unanswered. See Remark 3.3 in [2] for instances where the answer is affirmative. Next, we show that ZId(A) is, in fact, a normal coherent frame Yosida frame, and we describe its compact elements. A frame homomorphism h: L → M is codense if, for any a ∈ L, h(a) = 1 implies a = 1. Recall from [2, Lemma 1.2] that any codense image of a normal frame is normal. We shall write zA : RL(A) → ZId(A) for the frame homomorphism emanating from the nucleus constructed in the proof of Proposition 3.2. That is, _ [ zA(I) = {M(a) | a ∈ I} = {M(a) | a ∈ I}. ZId(A) Theorem 3.5. If A is an f-ring with bounded inversion, then ZId(A) is a normal coherent Yosida frame, and the set of its compact elements is k(ZId(A)) = {M(a) | a ∈ A} = {M(a) | a ∈ A+}. Proof. The equality of the last two sets above follows from the fact that M(a) = M(a2) for every a ∈ A, and squares are positive in any f-ring. Let I be a compact element in ZId(A). The equality _ I = {M(u) | u ∈ I} ZId(A) implies that there are finitely many elements u1, . . . , un in I such that 2 2 I = M(u1) ∨ · · · ∨ M(un) = M(u1) ∨ · · · ∨ M(un). + We claim that, for any finitely many a1, . . . , an in A ,

(†) M(a1) ∨ · · · ∨ M(an) = M(a1 + ··· + an). That the left hand side of (†) is contained in the right hand side follows from Lemma 3.1. To see the other inclusion, let J be a z-ideal containing each of the ideals M(ai). Then a1 + ··· + an ∈ J, so that M(a1 + ··· + an) ⊆ J since J is a z-ideal. This says M(a1 + ··· + an) ⊆ J is the least upper bound for the set {M(a1),...,M(an)} in the frame ZId(A), which is exactly what (†) says. Consequently, every compact element of ZId(A) is of the form M(a) 356 THEMBA DUBE for some a ∈ A. Next, let b ∈ A, and consider any directed family {Iα} of S elements of ZId(A) with M(b) ⊆ αIα. Then b ∈ Iα0 , for some index α0.

Since Iα0 is a z-ideal, M(b) ⊆ Iα0 . Therefore M(b) is compact. In all then, k(ZId(A)) = {M(a) | a ∈ A} = {M(a) | a ∈ A+}; and the ideals M(a), a ∈ A, generate ZId(A), showing that ZId(A) is an algebraic frame. That ZId(A) is compact follows easily from the fact that A has identity. Finally, from Lemma 3.1(a) we have that the meet of two compact elements is compact, hence ZId(A) is coherent. Since each M(a) is an intersection of maximal ideals of A, and maximal ideals of A are exactly the maximal elements of the frame ZId(A), it follows that ZId(A) is a Yosida frame. We prove its normality by showing that the surjective frame homomorphism zA : RL(A) → ZId(A) is codense. This will establish normality of ZId(A) because RL(A) is normal. Consider then any J ∈ RL(A) such that [ zA(J) = {M(u) | u ∈ J} = 1ZId(A) = A. There is therefore an element u ∈ J such that 1 ∈ M(u). But now this implies u is invertible, and so J = A = 1RL(A), hence ZId(A) is a codense image of a normal frame, which makes it normal. Remark 3.6. If we assume that the sum of two z-ideals in A is a z-ideal then an argument similar to that employed by Banaschewski [2] to show that L(A) is coherently normal can be modiﬁed, using M(a) in place of the principal `-ideal generated by a, to show that ZId(A) is coherently normal. We shall now identify the primes (i.e., the meet-irreducible elements) of ZId(A) with a view to characterizing those A for which ZId(A) is regular. Since ZId(A) is coherent, we know from Mart´ınezand Zenk [8, Theorem 2.4(a)] that ZId(A) is regular if and only if every prime element in this frame is minimal prime. So it behoves us to describe the primes of this frame. Lemma 3.7. If A is an f-ring with bounded inversion, then the prime elements of ZId(A) are precisely the prime z-ideals of A. Proof. Let P be meet-irreducible in ZId(A). We must show that P is a prime ideal in A. Consider ab ∈ P . Then M(a) ∩ M(b) = M(ab) ⊆ P since P is a z-ideal. Since P is meet-irreducible in ZId(A), we have M(a) ⊆ P or M(b) ⊆ P . Therefore a ∈ P or b ∈ P , and so P is a prime ideal in A. On the other hand, if P is a prime z-ideal in A, and I ∩ J ⊆ P for some I,J ∈ ZId(A), then IJ ⊆ I ∩ J ⊆ P , so that I ⊆ P or J ⊆ P by primeness, showing that P is meet-irreducible in ZId(A). Remark 3.8. For an f-ring A with bounded inversion we may deﬁne the Z- spectrum of A as the space of its prime z-ideals with the hull-kernel topology. By this lemma, it is then the frame-theoretic spectrum of ZId(A). A NOTE ON LATTICES OF z-IDEALS OF f-RINGS 357

Recall that a ring A is von Neumann regular if for every a ∈ A there exists b ∈ A such that a = a2b. It is well known that a semiprime ring is von Neumann regular if and only if every prime ideal in it is minimal prime. Since maximal ideals and minimal prime ideals are z-ideals, and since every prime ideal contains a minimal prime ideal, and is contained in a maximal ideal, we deduce from the theorem of Mart´ınezand Zenk cited above the following result. Corollary 3.9. If A is a semiprime f-ring with bounded inversion, then ZId(A) is regular if and only if A is von Neumann regular.

We now say more about the homomorphism zA : RL(A) → ZId(A). Let us recall from [2] that the compact elements of RL(A) are precisely the radicals of the principal `-ideals of positive elements of A. For any a ∈ A, write [a] = {x ∈ A | |x| ≤ s|a| for some s ≥ 0}, for the principal `-ideal√ of A generated by a. The radical of an ideal J will here be written as J. It is an `-ideal if J is an `-ideal. This is all in [2], and it is observed there that the set of compact elements of RL(A) is k(RL(A)) = {p[a] | a ∈ A+}. Let us note, for use in the upcoming proof, that if A has bounded inversion, then for any 0 ≤ a ≤ b, M(a) ⊆ M(b), and for any c ∈ A, M(c) = M(|c|).

Proposition 3.10. Let zA : RL(A) → ZId(A) be as above for an f-ring A with bounded inversion.

(a) zA is a codense coherent map. (b) zA is dense if and only if the Jacobson radical and the nilradical of A coincide. (c) zA is injective if and only if every radical `-ideal of A is a z-ideal. Proof. (a) We have already shown the codensity of this map. Turning to + p its coherence, we claim that, for any a ∈ A , zA( [a]) = M(a). Since p p p a ∈ [a], we immediately have M(a) ⊆ zA( [a]). Now let x ∈ zA( [a]). Then there is a non-negative integer n such that xn ∈ [a]; so, there is an s ∈ A+ such that |x|n = |xn| ≤ sa. Thus, M(x) = M(|x|) = M(|x|n) ⊆ M(sa) = M(s) ∩ M(a) ⊆ M(a), p so that zA( [a]) ⊆ M(a), and hence equality. This tells us that zA is coherent. (b) Since the zero of RL(A) is the nilradical, and the zero of ZId(√A) is the√ Jacobson radical, zA is dense if and√ only if the nucleus z sends 0 to 0. This in turn holds if and only if 0 = Jac(A). (c) Any frame homomorphism is injective if and only if its right adjoint is surjective. The right adjoint of zA is the inclusion map. The result follows from this. 358 THEMBA DUBE

For the next result we assume that the sum of two z-ideals is a z-ideal. So we are still dealing with f-rings that include√ function rings. Note that the binary join in RL(A) is given by I ∨ J = I + J. Recall that a frame homomorphism h: L → M is called closed precisely when, for every a, b ∈ L and any u ∈ M,

h(a) ≤ h(b) ∨ u =⇒ a ≤ b ∨ h∗(u). A closed frame homomorphism whose right adjoint preserves directed joins is called a proper map. Although we can obviate use of the following lemma in the proof of our intended result, we shall present it because it also brings to the fore a notewrothy fact about sums of z-ideals.

Lemma 3.11. If the sum of two z-ideals in any ring A is a z-ideal, then the sum of any collection of z-ideals of A is a z-ideal.

Proof. A straightforward induction shows that the sum of any ﬁnitely many z-ideals is a z-ideal. Now let {Iα} be a collection of z-ideals of A. Let P a ∈ α Iα, and pick ﬁnitely many indices α1, . . . , αn and elements aαi ∈ Iαi such that a = aα + ··· + aα . Since Iα + ··· + Iα is a z-ideal containing 1 n P1 n a, we have M(a) ⊆ Iα1 + ··· + Iαn ⊆ α Iα, as desired. Proposition 3.12. If the sum of two z-ideals in an f-ring A with bounded inversion is a z-ideal, then zA is a proper map. Furthermore, if the nilradical and the Jacobson radical of A coincide, then the right adjoint of zA is a frame homomorphism.

Proof. We shall abbreviate zA as z, so that z∗ abbreviates the right adjoint of zA. We show ﬁrst that z is a closed map. Consider I,J ∈ RL(A) and K ∈ ZId(A) such that z(I) ⊆ z(J) ∨ K in ZId(A). Let u ∈ I. Then _ M(u) ≤ z(I) ≤ {M(v) | v ∈ J} ∨ K, ZId(A) which, by compactness, implies there is a w ∈ J such that

M(u) ≤ M(w) ∨ K = M(w) + K in ZId(A). Since M(w) and K are z-ideals, M(u) + K is a z-ideal, by hypothesis, and hence a radical `-ideal in A, meaning that the join M(w)∨K in RL(A) is pM(w) + K = M(w)+K. Thus, with the join below calculated in RL(A), we have

u ∈ M(u) ≤ M(w) ∨ K ≤ J ∨ K = J ∨ z∗(K), which shows that I ≤ J ∨ z∗(K), whence zA is closed. Next, we show that z∗ preserves directed joins. In fact, we show that z∗ preserves all joins. So A NOTE ON LATTICES OF z-IDEALS OF f-RINGS 359 let {Iα} ⊆ ZId(A). Then, by Lemma 3.11, ! ! _ X X z∗ Iα = z∗ Iα = Iα ZId(A) α α s s X X _ = Iα = z∗(Iα) = z∗(Iα). α α RL(A)

Therefore zA is a proper map. The latter part of the proposition follows from Proposition 3.10(b) because then z∗ sends the zero element to the zero element.

4. Functoriality emanating from z-ideals In this last section we say a word about functoriality. Let BfRng denote the category whose objects are f-rings with bounded inversion, and whose morphisms are `-ring homomorphisms. We aim to show that the assignment A 7→ ZId(A) is the object part of a functor BfRng → CohFrm, and that the association A 7→ zA : RL → ZId is a natural transformation. We need a lemma.

Lemma 4.1. For any φ: A → B in BfRng, the map

τ : k(ZId(A)) → k(ZId(B)) given by M(a) 7→ M(φ(a)) for a ∈ A+ is a lattice homomorphism.

Proof. Since the zeros of these lattices are the Jacobson radicals of the rings in question, this map sends zero to zero. It also clearly sends the top element to the top element. By Lemma 3.1,

τM(a) ∧ M(b) = τM(ab) = M(φ(ab)) = M(φ(a)φ(b)) = M(φ(a)) ∧ M(φ(b)) = τM(a) ∧ τM(b).

Finally, as noted in (†) in the proof of Theorem 3.5, for any a, b ∈ A+, M(a∨b) = M(a+b). So, since φ sends positive elements to positive elements, we have

τM(a) ∨ M(b) = τM(a + b) = M(φ(a) + φ(b)) = M(φ(a)) ∨ M(φ(b)) = τM(a) ∨ τM(b).

Therefore τ is a lattice homomorphism. 360 THEMBA DUBE

As a consequence of this lemma, given φ: A → B in BfRng, there is a unique coherent map φ¯: ZId(A) → ZId(B) that extends φ (see [5, p. 64]). Explicitly, φ¯ maps as follows: _ [ φ¯(J) = {M(φ(a)) | a ∈ J} = {M(φ(a)) | a ∈ J}. ZId(B) Recall from [2] that RL: BfRng → CohFrm is a functor whose action on a morphism φ: A → B is q[ RL(φ)(J) = {[φ(a)] | a ∈ J}, so that for any a ∈ A+, RL(φ)(p[a]) = pφ(a). Proposition 4.2. The mappings A 7→ ZId(A) and φ 7→ φ¯ deﬁne a functor ZId: BfRng → CohFrm such that A 7→ zA : RL → ZId is a natural transformation. Proof. One checks routinely that

ZId(idA) = idZId(A) and ZId(ψφ) = ZId(ψ) ZId(φ) for any φ: A → B and ψ : B → C in BfRng. To prove that

A 7→ zA : RL → ZId is a natural transformation, we need to show that, for any φ: A → B in BfRng, the square

z RL(A) A / ZId(A)

RL(φ) ZId(φ)

z RL(B) B / ZId(B) commutes. Since RL(A) is generated by its compact elements, it suﬃces to show that ZId(φ)zA and zBRL(φ) agree on the compact elements of RL(A). Consider then p[a] for a ∈ A+. We observed in the proof of Proposition 3.10 p that zA( [a]) = M(a). Thus, p ZId(φ)zA( [a]) = ZId(φ) M(a) = M(φ(a)), and p p zBRL(φ)( [a]) = zB( [φ(a)]) = M(φ(a)), which proves the result. Acknowledgement. Thanks are due to the referee for some most welcome comments. A NOTE ON LATTICES OF z-IDEALS OF f-RINGS 361

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(Themba Dube) Department of Mathematical Sciences, University of South Africa, P. O. Box 392, 0003 Pretoria, South Africa. [email protected] This paper is available via http://nyjm.albany.edu/j/2016/22-16.html.